A -multigrade equation is a Diophantine equation of the form
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for , ..., , where and are -vectors. Multigrade identities remain valid if a constant is added to each element of and (Madachy 1979), so multigrades can always be put in a form where the minimum component of one of the vectors is 1.
Moessner and Gloden (1944) give a bevy of multigrade equations. Small-order examples are the (2, 3)-multigrade with and :
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the (3, 4)-multigrade with and :
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and the (4, 6)-multigrade with and :
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(Madachy 1979).
A spectacular example with and is given by and (Guy 1994), which has sums
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Rivera considers multigrade equations involving primes, consecutive primes, etc.
Analogous multigrade identities to Ramanujan's fourth power identity of form
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can also be given for third and fifth powers, the former being
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with , 2, 3, for any positive integer , and where
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and the one for fifth powers
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for , 3, 5, any positive integer , and where
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with a complex cube root of unity and and for both cases rational for arbitrary rationals and .
Multigrade sum-product identities as binary quadratic forms also exist for third, fourth, fifth powers. These are the second of the following pairs.
For third powers with ,
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for , 3, , and or for arbitrary , , , , , and .
For fourth powers with ,
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for , 4, , for arbitrary , , , .
For fifth powers with ,
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for , 2, 3, 4, 5, , (which are the same for fourth powers) for arbitrary , , , , and one for seventh powers that uses .
For seventh powers with ,
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for to 7, , , for arbitrary, , , , , (Piezas 2006).
A multigrade 5-parameter binary quadratic form identity exists for with , 2, 3, 5. Given arbitrary variables , , , , and defining and , then
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for , 2, 3, 5 (T. Piezas, pers. comm., Apr. 27, 2006).
Chernick (1937) gave a multigrade binary quadratic form parametrization to for , 4, 6 given by
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an equation which depends on finding solutions to .
Sinha (1966ab) gave a multigrade binary quadratic form parametrization to for , 3, 5, 7 given by
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which depended on solving the system for and 4 with and satisfying certain other conditions.
Sinha (1966ab), using a result of Letac, also gave a multigrade parametrization to for , 2, 4, 6, 8 given by
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where and . One nontrivial solution can be given by , , and Sinha and Smyth proved in 1990 that there are an infinite number of distinct nontrivial solutions.